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Chapter 4: Problem 57

Solve each equation. Check the solution. $$ -2(x-2)+5 x=10 $$

Short Answer

Expert verified
x = 2 is

Step by step solution

01

Distribute h2{Distribute the -2}

Multiply nhh{2}(x-2) by -2 to simplify the equation. This gives {-2(x-2)= nh{-2x+4}
02

Combine Like Terms

Combine the terms -2x and 5x to get {3x}
03

Isolate the Variable

Subtract 4 from both sides to get {-3x+3x-4= 6x}. Addtionally, divide by the coefficient 3 to get h{x=2}
04

Check the Solution

Substitute x = 2 back into the original equation and see if it satisfies the equality consideration {(x-3)+ 8x=8x} this results {(1)+nxn6=nx}}.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a core concept in algebra that helps simplify expressions and solve equations. It involves distributing a multiplier over terms inside parentheses. In the given exercise, we start with the equation \( -2(x - 2) + 5x = 10 \). To apply the distributive property, we multiply -2otag{x} otag{-2} inside the parentheses. This transforms the equation into \( -2x + 4 + 5x = 10 \). The idea here is to eliminate the parentheses so that we can easily combine like terms in the next step.
Combining Like Terms
Combining like terms is the next step after applying the distributive property. Like terms have the same variable raised to the same power. In this equation, we combine -2x and 5x. This looks like: \( -2x + 5x \) , which simplifies to \( 3x \). Now the equation becomes \( 3x + 4 = 10 \). By combining like terms, the equation is simpler and closer to being solved.
Isolating the Variable
To solve for the variable \( x \), we need to get it by itself on one side of the equation. This is known as isolating the variable. Starting with \( 3x + 4 = 10 \) , we subtract 4 from both sides to keep the equation balanced: \( 3x + 4 - 4 = 10 - 4 \), giving us \( 3x = 6 \). Next, we divide by 3:\( x = \frac{6}{3} \). This results in \( x = 2 \). Isolating the variable involves inverse operations, like addition/subtraction or multiplication/division, to simplify the equation step by step.
Checking the Solution
After solving for \( x \), always check your solution by substituting it back into the original equation to verify it satisfies the equality. For \( x = 2 \) , we plug it back into \( -2(x - 2) + 5x = 10 \) : \( -2(2 - 2) + 5(2) \), which simplifies as follows: \( -2(0) + 10 = 10 \). The left side equals the right side, validating that \( x = 2 \) is indeed correct. This step ensures no mistakes were made and confirms the solution's accuracy.

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Most popular questions from this chapter

Joyce Nemeth bought each of her seven nephews a gift, either a DVD of the movie The Blind Side or the latest Beyonc茅 CD. The DVD cost \(\$ 14.95\) and the CD cost \(\$ 16.88\), and she spent a total of \(\$ 114.30 .\) How many of each DVD and CD did she buy?

Without graphing, answer the following questions for each linear system. (a) Is the system inconsistent, are the equations dependent, or neither? (b) Is the graph a pair of intersecting lines, a pair of parallel lines, or one line? (c) Does the system have one solution, no solution, or an infinite number of solutions? $$ \begin{aligned} &2 x-y=4\\\ &y+4=2 x \end{aligned} $$

Solve each system by the elimination method. $$ \begin{aligned} &5 x+8 y=10\\\ &24 y=-15 x-10 \end{aligned} $$

If \(x\) units of a product cost \(C\) dollars to manufacture and earn revenue of \(R\) dollars, the value of \(x\) at which the expressions for \(C\) and \(R\) are equal is called the break-even quantity-the number of units that produce 0 profit. (a) find the break-even quantity, and (b) decide whether the product should be produced on the basis of whether it will earn a profit. (Profit\(=\)Revenue\(-\)cost) \(C=105 x+6000 ; R=255 x\) No more than 400 units can be sold.

Simplify. $$ (3 x-4 y)+(4 y-3 x) $$
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